Essential Spectrum of the Weighted Laplacian on Noncompact Manifolds and Applications

TL;DR

This paper provides upper bounds for the essential spectrum of the weighted Laplacian on noncompact manifolds, relating it to volume growth, and explores cases of equality, with applications to hypersurface curvature estimates.

Contribution

It introduces new upper estimates for the essential spectrum of weighted Laplacians based on volume growth conditions and identifies cases where these bounds are sharp.

Findings

Upper bounds for the essential spectrum depending on volume growth

Examples demonstrating equality in spectral estimates

Applications to curvature estimates of hypersurfaces

Abstract

We obtain upper estimates for the bottom (that is, greatest lower bound) of the essential spectrum of weighted Laplacian operator of a weighted manifold under assumptions of the volume growth of their geodesic balls and spheres. Furthermore, we find examples where the equality occurs in the estimates obtained. As a consequence, we give estimates for the weighted mean curvature of complete noncompact hypersurfaces into weighted manifolds.